[Paper Review] Propagating spinors on a tetrahedral spacetime lattice

Brendan Foster, Ted Jacobson|ArXiv.org|Oct 17, 2003

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This paper proposes a discrete path integral formulation for massless spinors on a tetrahedral spacetime lattice in 3+1 dimensions, using a hyperdiamond lattice structure with null faces to ensure causal propagation. The amplitude for a path with N steps and B bends is ±(1/2)^N (i/√3)^B, and the method yields the correct retarded propagator for the Weyl equation in the continuum limit, recovering Lorentz invariance and relativistic dynamics from a simple geometric rule without Grassmann variables.

ABSTRACT

We derive a discrete path integral for massless fermions on a hypercubic spacetime lattice with null faces. The amplitude for a path with N steps and B bends is +/- (1/2)^N (i/sqrt{3})^B.

Motivation & Objective

• To develop a discrete, non-Grassmannian path integral formulation for massless fermions in 3+1 dimensions that preserves key features of Feynman’s 1+1D checkerboard model.
• To construct a spacetime lattice—specifically a hyperdiamond lattice—where causal propagation and relativistic dynamics emerge from geometric rules.
• To ensure the discrete propagator converges to the correct continuum retarded propagator for the Weyl equation.
• To explore whether a simple, discrete dynamics can underlie the standard relativistic quantum field theory description of massless spinors.

Proposed method

• Uses a hyperdiamond lattice with edges aligned along four spacetime vectors n_i^μ = (1, α n̂_i), where the n̂_i are unit vectors pointing to the vertices of a regular tetrahedron.
• Sets the step speed α = 3 to satisfy the Courant condition and make the polyhedral faces null, ensuring causal propagation.
• Derives the amplitude for a path with N steps and B bends as ±(1/2)^N (i/√3)^B, with the i factor arising from geometric bends.
• Expresses the 4-vector σ^μ in terms of the lattice vectors and spin projection operators to connect the discrete structure to the Weyl equation.
• Demonstrates that the discrete path integral converges to the retarded propagator of the Weyl equation in the continuum limit.
• Analyzes the norm and unitarity of the evolution operator, showing norm loss and non-orthogonality in the discrete case, yet unitarity is recovered in the continuum.

Experimental results

Research questions

• RQ1 Can a simple, discrete path integral formulation for massless spinors be constructed in 3+1 dimensions that mirrors the elegance of Feynman’s 1+1D checkerboard?
• RQ2 Does a hyperdiamond lattice with null faces and tetrahedral symmetry support a consistent, convergent propagator for the Weyl equation?
• RQ3 How does the amplitude rule based on bends and step count reproduce the correct relativistic dynamics in the continuum limit?
• RQ4 Why does the discrete evolution violate unitarity while the continuum limit remains unitary?
• RQ5 What is the role of the tetrahedral symmetry and the specific choice α = 3 in ensuring causal structure and convergence?

Key findings

• The amplitude for a path with N steps and B bends is ±(1/2)^N (i/√3)^B, derived from geometric and symmetry considerations.
• With α = 3, the lattice faces become null, satisfying the Courant condition and ensuring causal propagation.
• The discrete path integral converges to the retarded propagator of the Weyl equation in the continuum limit, recovering Lorentz invariance.
• The evolution operator decreases the norm of a state by a factor of 1/2 per step and fails to preserve orthogonality, indicating discrete unitarity violation.
• Despite discrete unitarity violation, the continuum limit is unitary, suggesting that the loss of norm and increased overlap in evolutions cancel in the limit.
• The norm of the evolution matrix A is less than or equal to 1, with equality only when at least three of the spinor projection angles coincide.